WASHINGTON (Reuters) - Eric Cantor, the often combative second-ranking Republican in the U.S. House of Representatives, sought on Tuesday to rebrand himself and his party, voicing hope that they can work with President Barack Obama for the sake of all Americans.

While not wavering from his conservative principles and desire to tame the record U.S. debt, Cantor expressed a new eagerness to help the needy in such areas as education, healthcare, immigration and moving up the economic ladder.

"Over the next two years, the House (Republican) majority will pursue an agenda based on a shared vision of creating the conditions for health, happiness and prosperity for more Americans and their families," Cantor said in a speech to the American Enterprise Institute, a Washington think tank.

Some Democrats mocked Cantor's bid to "rebrand" his party, with House Minority Whip Steny Hoyer noting that it was at least Cantor's fourth time to do so.

But Charles Schumer, the Senate's No. 3 Democrat, hailed the House Republican leader's tone and message.

"If House Republicans can adapt their agenda to match Leader Cantor's words, this Congress could surprise people with how productive it can be," Schumer said.

The House Republican leader did not endorse immigration reforms backed by Obama but voiced an openness on the matter.

Cantor said he favored providing "an opportunity for legal residence and citizenship for those who were brought to this country as children and who know no other home."

That appeared to represent a reversal for Cantor, who in 2010 voted against the Dream Act, which would have cleared the way for such young people to remain in the United States.

Cantor gave little ground on any of the other differences between House Republicans and Obama in his speech, which his office billed as a major policy address. But he offered a marked change in tone and a new willingness to get things done on a number of fronts important to voters.

Republicans were hammered in the 2012 election, which saw Obama win a second term and Democrats gain seats in the House and Senate. The Republican Party was labeled by some critics as "The Party of No," one that preferred gridlock to compromise.

A Washington Post-ABC News/Washington poll last month found that 67 percent of Americans say that Republicans are doing "too little" to work with Obama.

The survey gave Republicans in Congress an approval rating of 24 percent, compared to a 37 percent rating for Democrats. Obama's approval recently hit a four-year high of 60 percent.

To be sure, differences remain.

As Cantor spoke, Republicans and Obama exchanged barbs in their latest standoff over deficit reduction, one likely to lead to $85 billion in damaging across-the-board budget cuts in March.

FOCUS ON 'WHAT LIES BEYOND'

Cantor made only passing reference to the bitter fights with Obama over "cliffs, debt ceilings and budgets" in which he has played such a visible role.

It is time, he said, to focus on "what lies beyond" them, including education, jobs, healthcare and innovation.

"It is my hope that I can stand before you in two years and report back that our side, as well as the president's, found within us the ability to set differences aside, to provide relief to so many millions of Americans who simply want their lives to work again," Cantor said.

Cantor, 49, from Virginia is widely seen as a possible successor to Ohio's John Boehner, 63, as House Speaker, the chamber's top job.

While the two insist that they have a close working relationship, at times they have offered competing visions. On Tuesday, however, Boehner said they were on the same page.

Speaking with reporters after a meeting with House Republicans, Boehner said, "As I told the members, Eric's giving a very important speech."

"If we're going to connect with the American people, it's important that they see, not only that we're serious about solving our debt problem, but we're serious about addressing issues like energy, like education, to show really the breadth of the effort that we're involved in," Boehner said.

During the past two years, the White House has tried to make Cantor the face of the unpopular Republican House, and has made it clear that Obama prefers working with Boehner.

In an apparent effort to present a softer personal image, Cantor punctuated his speech with references to a long lineup of people, including his wife and three children, his father, a Baltimore nurse and a police officer from his hometown of Richmond, Virginia.

He spoke of visiting an inner-city school this week and introduced a student from the school and his father to illustrate his interest in finding new solutions in education.